User:Halibutt/Spacetime/Curvature of time

Curvature of time

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Figure 5-3. Einstein's argument suggesting gravitational redshift

In the discussion of special relativity, forces played no more than a background role. Special relativity assumes the ability to define inertial frames that fill all of spacetime, all of whose clocks run at the same rate as the clock at the origin. Is this really possible? In a nonuniform gravitational field, experiment dictates that the answer is no. Gravitational fields make it impossible to construct a global inertial frame. In small enough regions of spacetime, local inertial frames are still possible. General relativity involves the systematic stitching together of these local frames into a more general picture of spacetime.

Shortly after the publication of the general theory in 1916, a number of scientists pointed out that general relativity predicts the existence of gravitational redshift. Einstein himself suggested the following thought experiment: (i) Assume that a tower of height h (Fig. 5‑3) has been constructed. (ii) Drop a particle of rest mass m from the top of the tower. It falls freely with acceleration g, reaching the ground with velocity v = (2gh)^1/2, so that its total energy E, as measured by an observer on the ground, is m = ½mv²/c² = m + mgh/c². (iii) A mass-energy converter transforms the total energy of the particle into a single high energy photon, which it directs upward. (iv) At the top of the tower, an energy-mass converter transforms the energy of the photon E' back into a particle of rest mass m'.

It must be that m = m', since otherwise one would be able to construct a perpetual motion device. We therefore predict that E' = m, so that

${\frac {E'}{E}}={\frac {h\nu \,'}{h\nu }}=$ ${\frac {m}{m+mgh/c^{2}}}=$ $1-{\frac {gh}{c^{2}}}$

A photon climbing in Earth's gravitational field loses energy and is redshifted. Early attempts to measure this redshift through astronomical observations were somewhat inconclusive, but definitive laboratory observations were performed by Pound & Rebka (1959) and later by Pound & Snider (1964).

Light has an associated frequency, and this frequency may be used to drive the workings of a clock. The gravitational redshift leads to an important conclusion about time itself: Gravity makes time run slower. Suppose we build two identical clocks whose rates are controlled by some stable atomic transition. Place one clock on top of the tower, while the other clock remains on the ground. An experimenter on top of the tower observes that signals from the ground clock are lower in frequency than those of the clock next to her on the tower. Light going up the tower is a just a wave, and it is impossible for wave crests to disappear on the way up. Exactly as many oscillations of light arrive at the top of the tower as were emitted at the bottom. The experimenter concludes that the ground clock is running slow, and can confirm this by bringing the tower clock down to compare side-by-side with the ground clock. For a 1 km tower, the discrepancy would amount to about 9.4 nanoseconds per day, easily measurable with modern instrumentation.

Clocks in a gravitational field do not all run at the same rate. Experiments such as the Pound–Rebka experiment have firmly established curvature of the time component of spacetime. The Pound–Rebka experiment says nothing about curvature of the space component of spacetime. But note that the theoretical arguments predicting gravitational time dilation do not depend on the details of general relativity at all. Any theory of gravity will predict gravitational time dilation if it respects the principle of equivalence. This includes Newtonian gravitation. A standard demonstration in general relativity is to show how, in the "Newtonian limit" (i.e. the particles are moving slowly, the gravitational field is weak, and the field is static), curvature of time alone is sufficient to derive Newton's law of gravity.

Newtonian gravitation is a theory of curved time. General relativity is a theory of curved time and curved space. Given G as the gravitational constant, M as the mass of a Newtonian star, and orbiting bodies of insignificant mass at distance r from the star, the spacetime interval for Newtonian gravitation is one for which only the time coefficient is variable:

$\Delta s^{2}=\left(1-{\frac {2GM}{c^{2}r}}\right)(c\Delta t)^{2}$${\displaystyle }$$-\,(\Delta x)^{2}-(\Delta y)^{2}-(\Delta z)^{2}$

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Curvature of space

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The $(1-2GM/(c^{2}r))$ coefficient in front of $(c\Delta t)^{2}$ describes the curvature of time in Newtonian gravitation, and this curvature completely accounts for all Newtonian gravitational effects. As expected, this correction factor is directly proportional to $G$ and $M$, and because of the $r$ in the denominator, the correction factor increases as one approaches the gravitating body, meaning that time is curved.

But general relativity is a theory of curved space and curved time, so if there are terms modifying the spatial components of the spacetime interval presented above, shouldn't their effects be seen on, say, planetary and satellite orbits due to curvature correction factors applied to the spatial terms?

The answer is that they are seen, but the effects are tiny. The reason is that planetary velocities are extremely small compared to the speed of light, so that for planets and satellites of the solar system, the $(c\Delta t)^{2}$ term dwarfs the spatial terms.

Despite the minuteness of the spatial terms, the first indications that something was wrong with Newtonian gravitation were discovered over a century-and-a-half ago. In 1859, Urbain Le Verrier, in an analysis of available timed observations of transits of Mercury over the Sun's disk from 1697 to 1848, reported that known physics could not explain the orbit of Mercury, unless there possibly existed a planet or asteroid belt within the orbit of Mercury. The perihelion of Mercury's orbit exhibited an excess rate of precession over that which could be explained by the tugs of the other planets. The ability to detect and accurately measure the minute value of this anomalous precession (only 43 arc seconds per tropical century) is testimony to the sophistication of 19th century astrometry.

Figure 5-4. General relativity is a theory of curved time and curved space. Click here to animate

As the famous astronomer who had earlier discovered the existence of Neptune "at the tip of his pen" by analyzing wobbles in the orbit of Uranus, Le Verrier's announcement triggered a two-decades long period of "Vulcan-mania", as professional and amateur astronomers alike hunted for the hypothetical new planet. This search included several false sightings of Vulcan. It was ultimately established that no such planet or asteroid belt existed.

In 1916, Einstein was to show that this anomalous precession of Mercury is explained by the spatial terms in the curvature of spacetime. Curvature in the temporal term, being simply an expression of Newtonian gravitation, has no part in explaining this anomalous precession. The success of his calculation was a powerful indication to Einstein's peers that the general theory of relativity could be correct.

The most spectacular of Einstein's predictions was his calculation that the curvature terms in the spatial components of the spacetime interval could be measured in the bending of light around a massive body. Light has a slope of ±1 on a spacetime diagram. Its movement in space is equal to its movement in time. For the weak field expression of the invariant interval, Einstein calculated an exactly equal but opposite sign curvature in its spatial components.

$\Delta s^{2}=\left(1-{\frac {2GM}{c^{2}r}}\right)(c\Delta t)^{2}$$-\,\left(1+{\frac {2GM}{c^{2}r}}\right)\left[(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}\right]$

In Newton's gravitation, the $(1-2GM/(c^{2}r))$ coefficient in front of $(c\Delta t)^{2}$ predicts bending of light around a star. In general relativity, the $(1+2GM/(c^{2}r))$ coefficient in front of $\left[(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}\right]$ predicts a doubling of the total bending.

The story of the 1919 Eddington eclipse expedition and Einstein's rise to fame is well told elsewhere.

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Sources of spacetime curvature

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Figure 5-5. Contravariant components of the stress–energy tensor.

In Newton's theory of gravitation, the only source of gravitational force is mass.

In contrast, general relativity identifies several sources of spacetime curvature in addition to mass. In the Einstein field equations, the sources of gravity are presented on the right-hand side in $T_{\mu \nu },$ the stress–energy tensor.

Fig. 5‑5 classifies the various sources of gravity in the stress-energy tensor:

• $T^{00}$ (red): The total mass-energy density, including any contributions to the potential energy from forces between the particles, as well as kinetic energy from random thermal motions.
• $T^{0i}$ and $T^{i0}$ (orange): These are momentum density terms. Even if there is no bulk motion, energy may be transmitted by heat conduction, and the conducted energy will carry momentum.
• $T^{ij}$ are the rates of flow of the i-component of momentum per unit area in the j-direction. Even if there is no bulk motion, random thermal motions of the particles will give rise to momentum flow, so the i = j terms (green) represent isotropic pressure, and the i ≠ j terms (blue) represent shear stresses.

One important conclusion to be derived from the equations is that, colloquially speaking, gravity itself creates gravity. Energy has mass. Even in Newtonian gravity, the gravitational field is associated with an energy, E = mgh, called the gravitational potential energy. In general relativity, the energy of the gravitational field feeds back into creation of the gravitational field. This makes the equations nonlinear and hard to solve in anything other than weak field cases. Numerical relativity is a branch of general relativity using numerical methods to solve and analyze problems, often employing supercomputers to study black holes, gravitational waves, neutron stars and other phenomena in the strong field regime.

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Energy-momentum

Figure 5-6. (left) Mass-energy warps spacetime. (right) Rotating mass-energy distributions with angular momentum J generate gravitomagnetic fields H

In special relativity, mass-energy is closely connected to momentum. As we have discussed earlier in the section on Energy and momentum, just as space and time are different aspects of a more comprehensive entity called spacetime, mass-energy and momentum are merely different aspects of a unified, four-dimensional quantity called four-momentum. In consequence, if mass-energy is a source of gravity, momentum must also be a source. The inclusion of momentum as a source of gravity leads to the prediction that moving or rotating masses can generate fields analogous to the magnetic fields generated by moving charges, a phenomenon known as gravitomagnetism.

Figure 5-7. Origin of gravitomagnetism.

It is well known that the force of magnetism can be deduced by applying the rules of special relativity to moving charges. (An eloquent demonstration of this was presented by Feynman in volume II, chapter 13–6 of his Lectures on Physics, available online.) Analogous logic can be used to demonstrate the origin of gravitomagnetism. In Fig. 5‑7a, two parallel, infinitely long streams of massive particles have equal and opposite velocities −v and +v relative to a test particle at rest and centered between the two. Because of the symmetry of the setup, the net force on the central particle is zero. Assume v << c so that velocities are simply additive. Fig. 5‑7b shows exactly the same setup, but in the frame of the upper stream. The test particle has a velocity of +v, and the bottom stream has a velocity of +2v. Since the physical situation has not changed, only the frame in which things are observed, the test particle should not be attracted towards either stream. But it is not at all clear that the forces exerted on the test particle are equal. (1) Since the bottom stream is moving faster than the top, each particle in the bottom stream has a larger mass energy than a particle in the top. (2) Because of Lorentz contraction, there are more particles per unit length in the bottom stream than in the top stream. (3) Another contribution to the active gravitational mass of the bottom stream comes from an additional pressure term which, at this point, we do not have sufficient background to discuss. All of these effects together would seemingly demand that the test particle be drawn towards the bottom stream.

File:Galaxies-AGN-Inner-Structure.svg

Figure 5-8. Relativistic jet.

The test particle is not drawn to the bottom stream because of a velocity-dependent force that serves to repel a particle that is moving in the same direction as the bottom stream. This velocity-dependent gravitational effect is gravitomagnetism.

Matter in motion through a gravitomagnetic field is hence subject to so-called frame-dragging effects analogous to electromagnetic induction. It has been proposed that such gravitomagnetic forces underlie the generation of the relativistic jets (Fig. 5‑8) ejected by some rotating supermassive black holes.

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Pressure and stress

Quantities that are directly related to energy and momentum should be sources of gravity as well, namely internal pressure and stress. Taken together, mass-energy, momentum, pressure and stress all serve as sources of gravity: Collectively, they are what tells spacetime how to curve.

General relativity predicts that pressure acts as a gravitational source with exactly the same strength as mass-energy density. The inclusion of pressure as a source of gravity leads to dramatic differences between the predictions of general relativity versus those of Newtonian gravitation. For example, the pressure term sets a maximum limit to the mass of a neutron star. The more massive a neutron star, the more pressure is required to support its weight against gravity. The increased pressure, however, adds to the gravity acting on the star's mass. Above a certain mass determined by the Tolman–Oppenheimer–Volkoff limit, the process becomes runaway and the neutron star collapses to a black hole.

The stress terms become highly significant when performing calculations such as hydrodynamic simulations of core-collapse supernovae.

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Experimental verification

These predictions for the roles of pressure, momentum and stress as sources of spacetime curvature are elegant and play an important role in theory. In regards to pressure, the early universe was radiation dominated, and it is highly unlikely that any of the relevant cosmological data (e.g. nucleosynthesis abundances, etc.) could be reproduced if pressure did not contribute to gravity, or if it did not have the same strength as a source of gravity as mass-energy. Likewise, the mathematical consistency of the Einstein field equations would be broken if the stress terms didn't contribute as a source of gravity.

All that is well and good, but are there any direct, quantitative experimental or observational measurements that confirm that these terms contribute to gravity with the correct strength?